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\section*{Categories and Sheaves}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include mathematicscontents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

This entry provides a hyperlinked index for the book

\begin{itemize}%
\item [[Masaki Kashiwara]], [[Pierre Schapira]],

\emph{Categories and Sheaves},

Grundlehren der Mathematischen Wissenschaften \textbf{332}, Springer (2006)



\end{itemize}
on basics of [[category theory]] and the foundations of [[homological algebra]] and [[abelian sheaf cohomology]].

See also the related lecture notes

\begin{itemize}%
\item [[Pierre Schapira]], \emph{Categories and homological algebra} (2011) (\href{http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf}{pdf})

\end{itemize}
\textbf{Summary}

The book discusses the theory of [[presheaf|presheaves]] and [[sheaf|sheaves]] with an eye towards their application in [[homological algebra]] and with an outlook on [[stack]]s.

A self-contained introduction of the basics of [[presheaf]]-categories with detailed discussion of [[representable functor]]s and the corresponding notions of [[limit]]s, [[colimit]]s, [[adjoint functor]]s and [[ind-object]]s forms the first third of the book.

The second part describes central concepts and tools of modern category-theoretic [[homological algebra]] in terms of [[derived category|derived]] [[triangulated category|triangulated categories]].

The last part merges these two threads in a discussion of [[sheaf|sheaves]] in general and [[abelian sheaf|abelian sheaves]] in particular. This provides the machinery for the consideration of [[abelian sheaf cohomology]] conceptually embedded into the general notion of [[cohomology]] and [[(infinity,1)-category of (infinity,1)-sheaves|higher stacks]], on which the last section provides an outlook.

The organization and emphasis of the book (for instance of the [[category of sheaves]] as a [[localization]] of the category of [[presheaf|presheaves]]) makes it a suitable 1-categorical preparation for the [[infinity-category|infinity-categorical]] discussion of sheaves in

\begin{itemize}%
\item J. Lurie, \emph{[[Higher Topos Theory]]}

\end{itemize}
and of triangulated categories, i.e. [[stable (infinity,1)-category|stable infinity-categories]], in

\begin{itemize}%
\item J. Lurie, \emph{[[Stable Infinity-Categories]]}

\end{itemize}
On the other hand, [[topos]]-theoretic aspects of the [[category of sheaves]] are not emphasized, here

\begin{itemize}%
\item Moerdijk-MacLane, \emph{[[Sheaves in Geometry and Logic]]}

\end{itemize}
is the natural complementary reading. In particular sections V and VII there are directly useful for supplementing the concept of [[geometric morphism]] and its relation to [[localization]].

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{1_the_language_of_categories}{1 The language of categories}\dotfill \pageref*{1_the_language_of_categories} \linebreak
\noindent\hyperlink{2_limits}{2 Limits}\dotfill \pageref*{2_limits} \linebreak
\noindent\hyperlink{3_filtrant_limits}{3 Filtrant Limits}\dotfill \pageref*{3_filtrant_limits} \linebreak
\noindent\hyperlink{4_tensor_categories}{4 Tensor Categories}\dotfill \pageref*{4_tensor_categories} \linebreak
\noindent\hyperlink{5_generators_and_representability}{5 Generators and Representability}\dotfill \pageref*{5_generators_and_representability} \linebreak
\noindent\hyperlink{6_indization_of_categories}{6 Indization of Categories}\dotfill \pageref*{6_indization_of_categories} \linebreak
\noindent\hyperlink{7_localization}{7 Localization}\dotfill \pageref*{7_localization} \linebreak
\noindent\hyperlink{8_additive_and_abelian_categories}{8 Additive and Abelian Categories}\dotfill \pageref*{8_additive_and_abelian_categories} \linebreak
\noindent\hyperlink{9_accesible_objects_and_injective_objects}{9 $\pi$-accesible Objects and $F$-injective Objects}\dotfill \pageref*{9_accesible_objects_and_injective_objects} \linebreak
\noindent\hyperlink{10_triangulated_categories}{10 Triangulated Categories}\dotfill \pageref*{10_triangulated_categories} \linebreak
\noindent\hyperlink{11_complexes_in_additive_categories}{11 Complexes in Additive Categories}\dotfill \pageref*{11_complexes_in_additive_categories} \linebreak
\noindent\hyperlink{12_complexes_in_abelian_categories}{12 Complexes in Abelian Categories}\dotfill \pageref*{12_complexes_in_abelian_categories} \linebreak
\noindent\hyperlink{13_derived_categories}{13 Derived Categories}\dotfill \pageref*{13_derived_categories} \linebreak
\noindent\hyperlink{14_unbounded_derived_categories}{14 Unbounded Derived Categories}\dotfill \pageref*{14_unbounded_derived_categories} \linebreak
\noindent\hyperlink{15_indization_and_derivation_of_abelian_categories}{15 Indization and Derivation of Abelian Categories}\dotfill \pageref*{15_indization_and_derivation_of_abelian_categories} \linebreak
\noindent\hyperlink{16_grothendieck_topologies}{16 Grothendieck Topologies}\dotfill \pageref*{16_grothendieck_topologies} \linebreak
\noindent\hyperlink{17_sheaves_on_grothendieck_topologies}{17 Sheaves on Grothendieck Topologies}\dotfill \pageref*{17_sheaves_on_grothendieck_topologies} \linebreak
\noindent\hyperlink{18_abelian_sheaves}{18 Abelian Sheaves}\dotfill \pageref*{18_abelian_sheaves} \linebreak
\noindent\hyperlink{19_stacks_and_twisted_sheaves}{19 Stacks and Twisted Sheaves}\dotfill \pageref*{19_stacks_and_twisted_sheaves} \linebreak


The following lists chapterwise linked lists of keywords to relevant and related existing entries, as far as they already exist.

For a pedagogical motivation of the general topic under consideration here see

\begin{itemize}%
\item [[motivation for sheaves, cohomology and higher stacks]]

\end{itemize}
\hypertarget{1_the_language_of_categories}{}\subsection*{{1 The language of categories}}\label{1_the_language_of_categories}

\begin{itemize}%
\item [[foundations]]

\begin{itemize}%
\item [[ETCS]]


\item [[Grothendieck universe]]



\end{itemize}

\item [[category]]

\begin{itemize}%
\item [[morphism]]

\begin{itemize}%
\item [[hom-set]]


\item [[identity morphism]]


\item [[epimorphism]]


\item [[monomorphism]]


\item [[inverse]]


\item [[isomorphism]]


\item [[endomorphism]]


\item [[automorphism]]



\end{itemize}

\item [[object]]

\begin{itemize}%
\item [[initial object]]


\item [[terminal object]]


\item [[zero object]]



\end{itemize}

\item [[small category]]


\item [[locally small category]]


\item [[finite category]]


\item [[opposite category]]


\item [[subcategory]]


\item general examples

\begin{itemize}%
\item [[discrete category]]


\item [[partial order|poset]]


\item [[groupoid]]


\item [[monoid]]


\item [[group]]


\item [[arrow category]]

\begin{itemize}%
\item [[comma category]]

\begin{itemize}%
\item [[category of elements]]


\item [[generalized universal bundle]]



\end{itemize}

\item [[subobject]]



\end{itemize}


\end{itemize}

\item [[partially ordered category]]


\item concrete examples

\begin{itemize}%
\item classical examples: sets with [[stuff, structure, property|extra structure]]

\begin{itemize}%
\item [[Set]]


\item [[Top]]


\item [[Vect]]


\item [[simplex category]]



\end{itemize}

\item [[point]]


\item [[globe]]


\item [[fundamental groupoid]]



\end{itemize}


\end{itemize}

\item [[functor]]

\begin{itemize}%
\item [[stuff, structure, property]]

\begin{itemize}%
\item [[forgetful functor]]


\item [[essentially surjective functor]]


\item [[full functor]]


\item [[faithful functor]]


\item [[subcategory]]



\end{itemize}

\item [[natural transformation]]

\begin{itemize}%
\item [[functor category]]


\item [[skeleton]]


\item [[equivalence|equivalence of categories]]



\end{itemize}

\item [[contravariant functor]]



\end{itemize}

\item [[presheaf]]

\begin{itemize}%
\item [[representable functor]]


\item [[Yoneda lemma]]

\begin{itemize}%
\item [[Yoneda embedding]]


\item [[Yoneda extension]]


\item [[free cocompletion]]


\item [[co-Yoneda lemma]]



\end{itemize}

\item [[adjoint functor]]

\begin{itemize}%
\item [[adjunction]]


\item [[examples of adjoint functors]]



\end{itemize}


\end{itemize}

\item functor and presheaf categories

\begin{itemize}%
\item [[simplicial set]]


\item [[presheaf on open subsets]]



\end{itemize}


\end{itemize}
\hypertarget{2_limits}{}\subsection*{{2 Limits}}\label{2_limits}

\begin{itemize}%
\item [[limit]]

\begin{itemize}%
\item [[colimit]]


\item examples

\begin{itemize}%
\item [[limits and colimits by example]]


\item [[product]]


\item [[coproduct]]


\item [[pullback]]


\item [[pushout]]


\item [[fiber product]]


\item [[equalizer]]


\item [[coequalizer]]


\item [[kernel]]


\item [[cokernel]]



\end{itemize}

\item [[weighted limit]]


\item [[continuous functor]]



\end{itemize}

\item [[Kan extension]]

\begin{itemize}%
\item [[Yoneda extension]]

\end{itemize}

\item [[cofinal functor]]


\item [[cofinally small category]]



\end{itemize}
\hypertarget{3_filtrant_limits}{}\subsection*{{3 Filtrant Limits}}\label{3_filtrant_limits}

\begin{itemize}%
\item [[filtered category]]


\item [[filtered limit]]


\item [[exact functor]]

\begin{itemize}%
\item [[flat functor]]

\end{itemize}


\end{itemize}
\hypertarget{4_tensor_categories}{}\subsection*{{4 Tensor Categories}}\label{4_tensor_categories}

\begin{itemize}%
\item [[monoidal category]]

\begin{itemize}%
\item [[braided monoidal category]]


\item [[closed monoidal category]]



\end{itemize}

\item [[enriched category theory]]

\begin{itemize}%
\item [[enriched category]]

\end{itemize}


\end{itemize}
\hypertarget{5_generators_and_representability}{}\subsection*{{5 Generators and Representability}}\label{5_generators_and_representability}

\begin{itemize}%
\item [[image]]

\begin{itemize}%
\item [[coimage]]


\item [[strict morphism]]

\begin{itemize}%
\item [[strict epimorphism]]


\item [[regular epimorphism]]



\end{itemize}


\end{itemize}

\item [[subobject]]


\item [[generator]]



\end{itemize}
\hypertarget{6_indization_of_categories}{}\subsection*{{6 Indization of Categories}}\label{6_indization_of_categories}

\begin{itemize}%
\item [[ind-object]]

\begin{itemize}%
\item [[pro-object]]


\item [[finitely presentable object]]


\item [[compact object]]



\end{itemize}


\end{itemize}
\hypertarget{7_localization}{}\subsection*{{7 Localization}}\label{7_localization}

\begin{itemize}%
\item [[homotopical category]]

\begin{itemize}%
\item [[category with weak equivalences]]


\item [[category of fibrant objects]]


\item [[Waldhausen category]]


\item [[model category]]



\end{itemize}

\item [[localization]]

\begin{itemize}%
\item [[calculus of fractions]]

\begin{itemize}%
\item [[multiplicative system]]


\item [[category of fractions]]



\end{itemize}

\item [[geometric embedding]]


\item [[homotopy category]]


\item [[derived functor]]



\end{itemize}


\end{itemize}
\hypertarget{8_additive_and_abelian_categories}{}\subsection*{{8 Additive and Abelian Categories}}\label{8_additive_and_abelian_categories}

\begin{itemize}%
\item [[group object]]


\item [[Ab]]


\item [[Ab-enriched category]]


\item [[additive and abelian categories]]

\begin{itemize}%
\item [[pre-additive category]]


\item [[additive category]]


\item [[pre-abelian category]]


\item [[abelian category]]


\item [[semi-abelian category]]


\item [[Grothendieck category]]



\end{itemize}

\item [[biproduct]]

\begin{itemize}%
\item [[direct sum]]


\item [[matrix calculus]]



\end{itemize}

\item [[chain complex]]

\begin{itemize}%
\item [[kernel]]


\item [[cokernel]]


\item [[homology]]



\end{itemize}


\end{itemize}
\hypertarget{9_accesible_objects_and_injective_objects}{}\subsection*{{9 $\pi$-accesible Objects and $F$-injective Objects}}\label{9_accesible_objects_and_injective_objects}

\hypertarget{10_triangulated_categories}{}\subsection*{{10 Triangulated Categories}}\label{10_triangulated_categories}

\begin{itemize}%
\item [[stable (infinity,1)-category]]


\item [[triangulated category]]


\item [[cohomological functor]]


\item [[null system]]



\end{itemize}
\hypertarget{11_complexes_in_additive_categories}{}\subsection*{{11 Complexes in Additive Categories}}\label{11_complexes_in_additive_categories}

\begin{itemize}%
\item [[homological algebra]]

\begin{itemize}%
\item [[category with translation]]


\item [[differential object]]

\begin{itemize}%
\item [[complex]]


\item [[double complex]]



\end{itemize}

\item [[mapping cone]]


\item [[category of chain complexes]]



\end{itemize}


\end{itemize}
\hypertarget{12_complexes_in_abelian_categories}{}\subsection*{{12 Complexes in Abelian Categories}}\label{12_complexes_in_abelian_categories}

\begin{itemize}%
\item [[snake lemma]]


\item [[quasi-isomorphism]]



\end{itemize}
\hypertarget{13_derived_categories}{}\subsection*{{13 Derived Categories}}\label{13_derived_categories}

\begin{itemize}%
\item [[derived category]]


\item [[derived functor on a derived category]]



\end{itemize}
\hypertarget{14_unbounded_derived_categories}{}\subsection*{{14 Unbounded Derived Categories}}\label{14_unbounded_derived_categories}

\begin{itemize}%
\item [[injective object]]


\item [[projective object]]



\end{itemize}
\hypertarget{15_indization_and_derivation_of_abelian_categories}{}\subsection*{{15 Indization and Derivation of Abelian Categories}}\label{15_indization_and_derivation_of_abelian_categories}

\hypertarget{16_grothendieck_topologies}{}\subsection*{{16 Grothendieck Topologies}}\label{16_grothendieck_topologies}

\begin{itemize}%
\item [[coverage]]

\begin{itemize}%
\item [[Grothendieck topology]]

\begin{itemize}%
\item [[Lawvere-Tierney topology]]

\end{itemize}

\item [[sieve]]


\item [[local epimorphism]]


\item [[local isomorphism]]


\item [[dense monomorphism]]



\end{itemize}

\item [[site]]



\end{itemize}
\hypertarget{17_sheaves_on_grothendieck_topologies}{}\subsection*{{17 Sheaves on Grothendieck Topologies}}\label{17_sheaves_on_grothendieck_topologies}

\begin{itemize}%
\item [[category of sheaves]]

\begin{itemize}%
\item [[sheaf]]


\item [[sheafification]]

\begin{itemize}%
\item [[geometric morphism]]


\item sheafification with respect to a [[Lawvere-Tierney topology]]


\item [[IPC-property]]



\end{itemize}

\item operations

\begin{itemize}%
\item [[direct image]]


\item [[inverse image]]


\item [[restriction and extension of sheaves]]



\end{itemize}


\end{itemize}


\end{itemize}
\hypertarget{18_abelian_sheaves}{}\subsection*{{18 Abelian Sheaves}}\label{18_abelian_sheaves}

\begin{itemize}%
\item hindsight motivation

\begin{itemize}%
\item [[simplicial presheaf|simplicial sheaf]]


\item [[Dold-Kan correspondence]]


\item [[abelian sheaf]]



\end{itemize}

\item [[closed monoidal structure on sheaves]]


\item [[ringed site]]


\item [[abelian sheaf cohomology]]

\begin{itemize}%
\item [[Deligne cohomology]]

\end{itemize}


\end{itemize}
\hypertarget{19_stacks_and_twisted_sheaves}{}\subsection*{{19 Stacks and Twisted Sheaves}}\label{19_stacks_and_twisted_sheaves}

\begin{itemize}%
\item [[higher category theory]]

\begin{itemize}%
\item [[2-category]]


\item [[2-groupoid]]



\end{itemize}

\item [[stack]]


\item outloook and further reading

\begin{itemize}%
\item [[infinity-stack homotopically]]

\begin{itemize}%
\item [[simplicial presheaf]]


\item [[model structure on simplicial presheaves]]


\item [[descent for simplicial presheaves]]



\end{itemize}

\item [[Higher Topos Theory]]

\begin{itemize}%
\item [[(infinity,1)-category of (infinity,1)-sheaves]]


\item [[descent]]



\end{itemize}


\end{itemize}


\end{itemize}
category: reference



\end{document}